## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 907

negative real axis respectively . PROOF . If N is a bounded normal operator then ,

by Corollary IX.3.15 , NN * - N * N = I if and only if tā = 1 for every spectral point ...

**positive**if and only if its spectrum lies on the unit circle , the real axis , or the non -negative real axis respectively . PROOF . If N is a bounded normal operator then ,

by Corollary IX.3.15 , NN * - N * N = I if and only if tā = 1 for every spectral point ...

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Q.E.D. Next we shall require some information on

transformations and their square roots . 2 LEMMA . A self adjoint transformation T

is

be the ...

Q.E.D. Next we shall require some information on

**positive**self adjointtransformations and their square roots . 2 LEMMA . A self adjoint transformation T

is

**positive**if and only if o ( T ) is a subset of the interval [ 0 , 00 ) . PROOF . Let Ebe the ...

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Let { Mis } be a

respect to a

defined by the equations Mijle ) S.am mij ( 2 ) u ( da ) , where e is any bounded

Borel ...

Let { Mis } be a

**positive**matrix measure whose elements Mis are continuous withrespect to a

**positive**o - finite measure kl . If the matrix of densities { mi ; } isdefined by the equations Mijle ) S.am mij ( 2 ) u ( da ) , where e is any bounded

Borel ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero